\title{ Re-Evaluation of the Low-Risk Anomaly \\ in Finance via Matching \\ \mbox{}
  \\{\large GOV2001 Final Paper}}

\author{Yang Lu \thanks{yang.lu2014@gmail.com}, Daniel Wu
  \thanks{danielwu@fas.harvard.edu}, Kwok
  Yu\thanks{kwok\_yu@harvard.edu}}

\date{\today}

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\begin{abstract}
The long-term success of low-risk stocks over high-risk stocks runs
contrary to the basic finance principle that risk is compensated with
higher expected returns. Our paper examines this low-risk anomaly
using Coarsened Exact Matching to balance high and low-risk stock
portfolios on industry, company size, and trading volume. After
matching, we find that the low-risk anomaly still exists but has a
more muted effect than in previous studies, especially when beta is
used as a measure of risk. We also find moments in which the low-risk
anomaly does not hold, most notably during the dot-com bubble. To
our knowledge, we are the first to apply matching techniques to the
study of the low-risk anomaly, and our findings complicate various
previous explanations of this phenomenon.
\end{abstract}

\section{Introduction}

The low-risk anomaly is one of the longest-standing puzzles in
finance. Contrary to basic principles of finance, low-risk stocks have
long outperformed high-risk stocks.  Risk of a stock or portfolio is
typically measured using volatility (standard deviation of returns) or
beta (slope of regression line between asset returns and market risk
premium).\footnote{Market risk premium = expected market returns -
risk-free rate.}  According to the Modern Portfolio Theory, market
participants are assumed to be risk averse and investing is a tradeoff
between risk and return. Thus, assets with a higher expected return
are generally riskier.  Similarly, the Sharpe-Lintner Capital Asset
Pricing Model (CAPM) that is commonly used to price securities assumes
a positive correlation between risk and return. However, historical
stock returns show that both high-volatility and high-beta stocks
consistently underperform low-volatility and low-beta stocks.

Behaviorial finance models have offered two primary explanations of
the low-risk anomaly: irrational market participants and limits to
arbitrage. First, irrational investors who exhibit overconfidence,
representativeness or extrapolation biases, or those who have a
preference for lotteries can drive up demand and prices for high-risk
stocks. Second, institutional benchmarks and leverage constraints
discourage investment in low-risk stocks and can lead fund managers to
exacerbate the anomaly rather than arbitrage it away.

Additionally, some have questioned the assumptions of CAPM
\citep{fama, jensen} while others argue that problems of sample bias
and spurious correlations from data mining are responsible for the
appearance of an anomaly \citep{bernstein}.

We contribute to the literature methodologically by introducing the
use of matching techniques to this particular topic. Using Coarsened
Exact Matching (CEM) \citep{cem}, we match low-risk portfolios with
high-risk portfolios on industry, company size, and trading volume
to create a more balanced data set in an attempt to isolate the
relationship between risk and return. Replicating results from the
influential piece titled \emph{Benchmarks as Limits to Arbitrage:
Understanding the Low-Volatility Anomaly} \citep{lowvol}, we use
monthly stock-level data from 1968 through 2008 to demonstrate the
existence of the low-risk anomaly.\footnote{See Appendix for the
replication results.} We then pre-processed the lowest and
highest-risk portfolios with CEM, improving the balance by almost 10\%,
and re-calculated the returns on the pruned portfolios.  Although the
low-risk anomaly still exists after matching, we find that the
spread of cumulative returns between the two portfolios narrows,
especially when using beta as a measure of risk.

Our work ultimately builds on the work of others in the
field \citep{lowvol, blitzetal, vlietetal}, who have contributed
methodologically by subjecting the low-risk anomaly to further
controls informed by theory. In contrast to others' work, we found a
moment, during the dot-com bubble, in which the low-risk anomaly
disappears. During this period, instead of finding that high-risk
stocks underperform, we find the opposite. This complicates the
existing propositions by scholars including those from \citep{fama,
lowvol}. Our study opens up questions about the effect of certain time
periods on the low-risk anomaly (e.g. October 1999 through June 2000)
and sheds some light on the muddy relationship between risk and
return, ultimately suggesting to scholars the possibility of
alternative explanations to the anomaly.

\section{Data}

The data set used in the paper comes from the Center for Research in
Security Prices (CRSP). It includes all stocks traded in the United
States from January 1968 through December 2008 on a monthly basis. The
stock-level data include the month-end close price, the total monthly
return, the monthly trading volume, the number of outstanding shares
as well as the major industry group to which a stock belongs
based on the Standard Industrial Classification (SIC) codes available
on CRSP.

We calculated the volatility and beta for each stock in the data set
using up to 60 months of trailing data (the data set thus went back to
January 1963). Volatility was calculated by taking the standard
deviation of returns for each stock. Beta was calculated by regressing
stock returns on excess market returns over the risk-free rate. Any
stock that did not have at least 24 months of data within the 60-month
trailing window was removed for that particular month. Market returns
and risk-free rates were obtained from Kenneth French's data library
at
Dartmouth.\footnote{See \url{http://mba.tuck.dartmouth.edu/pages/faculty/ken.
french/data_library.html} for more information.}


\section{Methodology}

We used matching techniques to re-evaluate the low volatility and low
beta anomalies. Matching is typically used in social science research
to improve balance between treatment and control groups by eliminating
observations that do not have a counterpart in the comparison group
\citep{rubina, rubinb, hoetal}. However, we believe that it can also
be useful to pre-process and prune two stock portfolios so that the
relationship between returns and risk can be more easily isolated. For
example, in a given month, if the lowest volatility portfolio contains
a handful of mega-cap, high-volume oil companies, but no such
companies exist in the highest volatility portfolio, those stocks can
be removed from the study to ensure balance between the two
portfolios.  It is important to note that removing stocks from the
study does not bias portfolio return estimates if they are removed
based on confounders (major industry groups, company size, trading
volume) or ``treatment assignment'' (volatility or beta) but not on
outcomes (stock returns).

Matching has also been shown to reduce model dependence \citep{hoetal,
cem}. In the context of this study, improving balance between the
highest and lowest volatility or beta portfolios could reduce a
portfolio's over-reliance on particular industries or types of stocks
in generating gains or losses. Although matching does not allow us to
prove causality in our study, we use these techniques in an attempt to
gain some additional insight into the relationship between risk and
returns.

Various matching techniques were considered in the study include
nearest neighbor propensity score matching (PSM), nearest neighbor
Mahalanobis distance matching, and CEM. We chose CEM because it
provides the best balance (as measured by the L1 statistic) between
the two portfolios.  Balance or the similarity of stock features is
our primary concern as we aim to re-evaluate the low-risk anomaly by
controlling for the background covariates of the portfolios.

\section{Empirical Analysis}

For each of the 492 months in the data set, stocks were sorted by
their trailing volatility and assigned to one of five volatility
portfolios (low to high) weighted by market capitalization. The same
procedure was followed to create five beta portfolios (low to
high). We ignored transaction costs when rebalancing the portfolios
each month.

We then used the CEM algorithm to match stocks in the lowest
volatility or beta portfolio (quintile 1) with stocks in the highest
volatility or beta portfolio (quintile 5) each month based on major
industry group, company size, and trading volume.

\begin{itemize}
%%  \setlength{\itemindent}{2.5cm}
\item {\textbf{Industry Group:}} The classification into various
  industry groups is based on the SIC code. Each stock is assigned to
  one of 83 major industry groups based on the SIC codes. Matching on
  a stock to which it belongs is important in that different sectors
  may have different trends and are susceptible to various economic
  conditions. Imagine an average stock in the mining sector would
  behave differently from one in the agriculture sector.
  
\item {\textbf{Trading Volume:}} Trading volume reflects the liquidity
  of a stock in the market. The higher the volume, the easier the
  stock is to be purchased. Liquidity can be a factor in the
  risk-return relationship. Trading volume is measured using the natural
  logarithm of trading volume for the prior month and is coarsened
  into quartiles.  Matching on volume quartiles prevents comparing
  stocks of very different liquidity and helps to see whether other
  factors exist that contribute to the aforementioned anomaly.

\item {\textbf{Company Size:}} Company size is defined by its market
  capitalization, which is the product of price per share and the
  number of outstanding shares. Company size is measured using the
  natural logarithm of market capitalization for the prior month and
  is coarsened into quartiles. Stocks of different sizes tend to
  behave differently. Large caps are followed by more analysts and
  institutions; thus chances of mispricing are lower. Matching on size
  quartiles is important in that it minimizes the chances of comparing
  stocks with vastly different degrees of price efficiency.
\end{itemize}

\clearpage

\begin{figure}%[htp]
\caption{\label{figure:matched} Post-Matching Returns by Volatility and Beta Quintile,
  Jan. 1968-Dec. 2008.} 
\centering 
\subfigure[Panel A]{
  \label{figure:matchvol}
  \includegraphics[width=0.5\textwidth]{matchplotvol}}

\subfigure[Panel B]{
  \label{figure:matchbeta}
  \includegraphics[width=0.5\textwidth]{matchplotbeta}}
\centering

\caption*{\scriptsize Notes: For each month, we use Coarsened Exact
  Matching to match stocks in the lowest volatility/beta portfolio to
  stocks in the highest volatility/beta portfolio based on major
  industry group, company size, and trading volume. Each stock is
  assigned to one of 83 major industry groups based on the SIC
  codes. Company size is measured using the natural logarithm of
  market capitalization for the prior month and is coarsened into
  quartiles. Trading volume is measured using the natural logarithm of
  trading volume for the prior month and is coarsened into
  quartiles. In January 1968, \$1 is invested into each portfolio
  according to capitalization weights and matching weights, and the
  cumulative value of each portfolio is calculated each month until
  December 2008. We estimated volatility and beta by using up to 60
  months of trailing returns (i.e. return data starting as early as
  January 1963) obtained from CRSP. At the end of each month, we
  rebalanced each portfolio, excluding all transaction costs. The
  original (pre-matched) returns shown in Figure \ref{figure:matched}
  are represented with dotted lines for comparison purposes.}

\end{figure}

Table \ref{tab:ret} details the various statistics of both the
original and the matched portfolios from January 1968 through December
2008. In Table \ref{tab:ret} and Panel A of
Figure \ref{figure:matched}, one dollar invested in the matched lowest
volatility portfolio in January 1968 grows to \$41.86 by December
2008. One dollar invested in the matched highest volatility portfolio
during the same time period was worth \$0.21 by December 2008. As a
comparison, the original lowest and highest volatility quintile
portfolios were worth \$62.72 and \$0.40, respectively. Thus, the low
volatility anomaly still exists even after matching on major industry
group, market cap, and trading volume. However, in many months, the
spread between the lowest and highest volatility portfolios has been
reduced compared to the original quintile portfolios.

In Panel B of Figure \ref{figure:matched}, one dollar invested in the
matched lowest beta portfolio in January 1968 grows to \$22.99 by
December 2008. One dollar invested in the matched highest volatility
portfolio during the same time period was worth \$7.08 by December
2008. As a comparison, the pre-matched lowest and highest beta
portfolios were worth \$59.84 and \$6.25, respectively. Thus, the
spread between the lowest and highest beta portfolios shrinks
dramatically after matching, although the lowest beta portfolio still
produces a superior return. What is even more noteworthy is that
during the period from October 1999 through June 2000, the matched
quintile 5 portfolio outperformed the matched quintile 1 portfolio.

\clearpage
\begin{landscape}

\begin{table}\small
\caption{\label{tab:ret} Returns by Volatility and Beta Quintile, Original vs. Matched, January 1968-December 2008}
\centering
\begin{tabular}{p{4.5cm} m{1.5cm} m{1.5cm} m{1.5cm}m{1.5cm} m{1.5cm} m{1.5cm}m{1.5cm} m{1.5cm} }
\hline

\mbox{} & \multicolumn{4}{c}{Volatility} & \multicolumn{4}{c}{Beta} \\ \hline

 & \multicolumn{2}{c}{Quintile 1} & \multicolumn{2}{c}{Quintile 5} & \multicolumn{2}{c}{Quintile 1} & \multicolumn{2}{c}{Quintile 5} \\ 
  \hline
& Original & Matched & Original & Matched & Original & Matched & Original & Matched \\
Cumulative value in 2008 of \$1 invested in 1968 & \$62.72 & \$41.86 & \$0.40 & \$0.21 & \$59.84 & \$22.99 & \$6.25 & \$7.08 \\ 
Geometric Average R_p - R_f & 4.43\% & 3.43\% & -7.99\% & -9.57\% & 4.32\% & 2.00\% & -1.25\% & -0.94\% \\ 
Average R_p - R_f & 5.29\% & 4.93\% & -2.74\% & -3.84\% & 5.06\% & 3.51\% & 2.54\% & 2.84\% \\ 
Standard Deviation & 13.09\% & 17.32\% & 31.86\% & 33.33\% & 12.20\% & 17.54\% & 27.16\% & 27.15\% \\ 
Sharpe Ratio & 0.40 & 0.28 & -0.09 & -0.12 & 0.42 & 0.20 & 0.09 & 0.10 \\ 

&&&& &&&&\\

Average R_p - R_m & 1.11\% & 0.75\% & -6.93\% & -8.02\% & 0.88\% & -0.67\% & -1.64\% & -1.34\% \\ 
Tracking Error & 7.02\% & 9.67\% & 20.37\% & 22.0\% & 10.15\% & 11.37\% & 14.08\% & 14.63\% \\ 
Information Ratio & 0.16 & 0.08 & -0.34 & -0.36 & 0.09 & -0.06 & -0.12 & -0.09 \\ 
Beta & 0.75 & 0.91 & 1.68 & 1.73 & 0.60& 0.85 & 1.56 & 1.53 \\ 
Alpha & 2.17 & 1.11 & -9.77 & -11.09 & 2.57 & 1.11 & -4.00 & -11.09 \\ 
   \hline
\end{tabular}


\caption*{\small 
               Notes: For each month, we formed matched portfolios based on
the original quintile 1 and 5 portfolios, respectively. Stocks in all
portfolios are cap-weighted. We estimated volatility and beta by using
up to 60 months of trailing returns (i.e., return data starting as
early as January 1963). The return on the market, $R_m$, and the
risk-free rate, $R_f$, are from Ken French's data library at
Dartmouth. The information ratio uses the market return for the
relevant universe, all stocks in the first five columns and the top
1,000 stocks in the last five columns. Average returns are monthly
averages multiplied by 12. Standard deviation and tracking error are
monthly standard deviations multiplied by the square root of 12.}

\end{table}

\end{landscape}



\begin{table} \footnotesize
\caption{\label{tab:balance} Monthly Imbalance and Portfolio Size, January 1968-December 2008}
\centering
\begin{tabular}{p{4.5cm} m{1.5cm} m{1.5cm} m{1.5cm} }

\hline

\mbox {} & Mean L1 & Std. Dev. of L1 & Mean N & Std. Dev. of N \\ \hline

Original (Volatility) & 0.982 & 0.007 & 1792.5 & 754.5 \\ Matched
(Volatility) & 0.822 & 0.080 & 480.0 & 308.7 \\ &&&& \\

Original (Beta) & 0.863 & 0.049 & 1781.7 & 765.9 \\
Matched (Beta) & 0.755 & 0.082 & 941.0 & 582.8 \\ \hline

\end{tabular}

\caption*{\small Notes: For each month from January 1968 to December 2008, we
  calculate L1 as a measure imbalance between the lowest and highest
  volatility/beta portfolios. An L1 value of 1 indicates total
  imbalance, with lower L1 values indicating less imbalance. The mean
  and standard deviation of the monthly L1 are displayed for the
  original data set and the matched data set for both volatility and
  beta. The mean and standard deviation of the monthly sample size (N)
  is also shown.}

\end{table}

As shown in Table \ref{tab:balance}, CEM matching reduces imbalance
from a monthly mean L1 value of 0.982 (1 indicates total imbalance) in
the original volatility data set to 0.822 in the matched volatility
data set. As a consequence of matching, the mean monthly sample size
was reduced from 1792.5 to 480.

Similarly, matching reduces imbalance from a monthly mean L1 value of
0.863 in the original beta data set to 0.755 in the matched beta data
set. As a consequence of matching, the mean monthly sample size was
reduced from 1781.7 to 941.

\section{Discussion and Conclusion}

To our knowledge, this study is the first of its kind in applying
matching techniques to analyze and evaluate the low-risk anomaly,
arguably one of the most interesting puzzles in finance.
Matching, especially CEM used in the study, allows us to
control for features other than risk before forming portfolios. 

Both sets of results largely confirm the existence of the low-risk
anomaly in a more rigorous statistical setting. However, the reduced
spread of cumulative returns between low and high-risk portfolios
after matching shows that at least a small portion of the low-risk
anomaly can be explained by the fact that stocks in the original
quintile 1 and quintile 5 portfolios did not share similar features
(i.e. major industry group, company size, and trading volume).
Interestingly, the magnitude of the anomaly was reduced more dramatically when
beta was used as a measure of risk rather than volatility. The reason behind this
phenomenon requires further research.

As mentioned, various matching techniques were considered, and CEM was
selected because it gave us the portfolios with the best
balance. However, we observed similar results when applying other
matching techniques (including PSM and Mahalanobis distance
matching).

We made two assumptions in the study, each of which could be relaxed
to further explore the issue. The first assumption is the absence of
the omitted variable bias. We matched stocks in quintile 1 with those
in quintile 5, or vice versa, based on industry group, company size,
and trading volume. We believe that these three variables best capture
the most fundamental feature of a stock and the impact of adding other
matching variables would be marginal at best.

We also assumed that volatility or beta of a stock is independent of
its industry group, company size, and trading volume. By matching on
the three variables, we assumed that the differences in both monthly
return and cumulative return between the original and the matched
portfolios are due to volatility or beta. Any potentially correlations
between volatility or beta and any of the three variables are assumed
to be zero.

Finally, the crossing-over of cumulative returns of the
matched quintile 5 beta portfolio and the matched quintile 1
beta portfolio, which was not observed in previous studies,
is interesting and worthy of further investigation. During the
dot-com bubble when irrational exuberance ruled the market, one
would expect the effect of the anomaly be to greater in magnitude, yet the contrary
was observed. This finding complicates the existing explanations of the
anomaly. From the perspective of an efficient market, one has
to explain why the effect of certain risk factors and cognitive biases
were not as strong during that period. This short
disappearance of the low-risk anomaly opens up questions about the
overwhelming effect of certain time periods.

Overall, this study re-confirms the existence of the low-risk anomaly
with added statistical rigor but raises new questions about the relationship
between risk and return. Our research also serves a starting point for scholars
to further explore the low-risk anomaly in extraordinary circumstances
such as the dot-com bubble.

\clearpage

\section{Appendix}

\section*{Replication}

Similar to \cite{lowvol}, we tracked stock returns from January 1968
to December 2008 using data from CRSP. For each month, we calculated
the volatility and beta for each stock in the data set using up to
60-months of trailing data (the data set thus went back to January
1963). Volatility was calculated by taking the standard deviation of
returns for each stock. Beta was calculated by regressing stock
returns on excess market returns over the risk-free rate (market
return minus the risk-free rate). Any stock that did not have at least
24-months of data within the 60-month trailing window was removed for
that particular month. Market returns and risk-free rates were
obtained from Ken French's data library at Dartmouth. For each of the
492 months in the study, stocks were sorted by their trailing
volatility and assigned to one of five volatility portfolios (low to
high) weighted by market capitalization. The same procedure was
followed to create five beta portfolios (low to high). We ignored
transaction costs when rebalancing the portfolios each month.  We also
conducted the same exercise while restricting the CRSP data to the top
1,000 stocks by market capitalization. The results from both studies
appear in Figure \ref{figure:plotMulti}.


\begin{figure}%[htp]
\centering      
\subfigure[Panel A]{
    \label{figure:plotA}
        \includegraphics[width=0.5\textwidth]{plotA}}

\subfigure[Panel B]{
        \label{figure:plotB}
        \includegraphics[width=0.5\textwidth]{plotB}}
\centering
\end{figure}

\begin{figure}
\centering
\subfigure[Panel C]{
       \label{figure:plotC}
        \includegraphics[width=0.5\textwidth]{plotC}}

\subfigure[Panel D]{
        \label{figure:plotD}
        \includegraphics[width=0.5\textwidth]{plotD}}

\centering

\caption{\label{figure:plotMulti} For each month, we sort all publicly traded stocks (Panels A and C) and the top 1,000 stocks by market capitalization (Panels B and D) tracked by CRSP (with at least 24 months of return history) into five equal quintiles according to trailing volatility (standard deviation) and beta. In January 1968, \$1 is invested into each portfolio according to capitalization weights. We estimate volatility and beta by using up to 60 months of trailing returns (i.e. return data starting as early as January 1963). At the end of each month, we rebalance each portfolio, excluding all transaction costs.}

\end{figure}


\clearpage

\begin{landscape}
\begin{table} \footnotesize

\begin{tabular}{p{4.5cm} m{1.5cm} m{1.5cm} m{1.5cm} m{1.5cm} m{1.5cm}  m{1.5cm} m{1.5cm} m{1.5cm} m{1.5cm} m{1.5cm}}

\hline

\mbox {} & \multicolumn{5}{c}{All Stocks} & \multicolumn{5}{c}{Top 1,000 Stocks} \\ \hline
\mbox {} & Low & Quintile 2 & Quintile 3 & Quintile 4 & High & Low & Quintile 2 & Quintile 3 & Quintile 4 & High \\ \hline

\bf{A.Volatility Sorts} & &&&&&&&&& \\

Cumulative value in 2008 of \$1 invested in 1968 & \$62.72 & \$40.54 & \$29.32 & \$11.60 & \$0.40 & \$58.70 & \$45.02 & \$26.93 & \$27.44 & \$5.86 \\

Geometric Average R_p - R_f & 4.43\% & 3.35\% & 2.55\% & 0.26\% & -7.99\% & 4.27\% & 3.61\% & 2.34\% & 2.39\% & -1.40\% \\

Average R_p - R_f & 5.29\% & 4.79\% & 4.92\% & 4.02\% & -2.74\% & 5.08\% & 4.79\% & 3.93\% & 4.66\% & 2.42\% \\

Standard Deviation & 13.09\% &16.75\% & 21.44\% & 27.04\% & 31.86\% & 12.72\% & 15.26\% & 17.54\% & 21.00\% & 27.28\% \\

Sharpe Ratio & 0.40 & 0.29 & 0.23 & 0.15 & -0.09 & 0.40 & 0.31 & 0.22 & 0.22 & 0.09 \\

& &&&&&&&&& \\

Average R_p - R_m & 1.11\% & 0.61\% & 0.74\% & -0.16\% & -6.93\% & 0.90\% & 0.61\% & -0.26\% & 0.48\% & -1.76\% \\
Tracking Error	& 7.02\%&	4.97\%&	8.25\%&	14.58\%&	20.37\%&	8.12\%&	6.08\%&	4.84\%&	8.35\%	&14.87\% \\ 
Information Ratio	& 0.16&	0.12&	0.09&	-0.01&	-0.34&	0.11&	0.10&	-0.05	&0.06&	-0.12 \\
Beta &	0.75&	1.01&	1.28&	1.53&	1.68&	0.70&	0.89&	1.06&	1.24&	1.53 \\
Alpha &	2.17&	0.57&	-0.42&	-2.36&	-9.77&	2.17&	1.06&	-0.52	&-0.51&	-3.98 \\ \hline

& &&&&&&&&& \\
\mbox {} & \multicolumn{5}{c}{All Stocks} & \multicolumn{5}{c}{Top 1,000 Stocks} \\ \hline
\mbox {} & Low & Quintile 2 & Quintile 3 & Quintile 4 & High & Low & Quintile 2 & Quintile 3 & Quintile 4 & High \\ \hline

\bf{B.Beta Sorts} & &&&&&&&&& \\

Cumulative value in 2008 of \$1 invested in 1968 & \$59.84 & \$57.14& \$33.39 & \$16.14 & \$6.25 & \$72.61 & \$48.85 & \$36.12 & \$19.70 &\$5.68  \\
Geometric Average R_p - R_f &	4.32\%&	4.20\%&	2.87\%&	1.08\%&	-1.25\%&	4.80\%&	3.81\%&	3.07\%&	1.57\%&	-1.48\% \\
Average R_p - R_f &	5.06\%&	5.11\%&	4.24\%&	3.14\%&	2.54\%&	5.56\%&	4.82\%&	4.41\%&	3.46\%&	1.89\%\\
Standard Deviation &	12.20\%&	13.47\%& 16.39\%&	19.97\%&	27.16\%& 12.40\%& 14.15\%& 16.24\%& 19.18\%& 25.58\%\\
Sharpe Ratio&	0.42 &	0.38 &	0.26 &	0.16 &	0.09 &	0.45 &	0.34 &	0.27 &	0.18 &	0.07 \\
	
& &&&&&&&&& \\									
Average R_p - R_m &	0.88\%&	0.93\%&	0.06\%&	-1.05\%&	-1.64\%&	1.38\%&	0.64\%&	0.23\%&	-0.72\%&	-2.29\% \\
Tracking Error	& 10.15\%	&7.40\%&	5.66\%&	6.32\%&	14.08\%&	9.65\%&	6.90\%&	5.09\%&	5.53\%&	12.34\% \\
Information Ratio &	0.09&	0.13&	0.01&	-0.17&	-0.12&	0.14&	0.09&	0.04&	-0.13&	-0.19\\ 
Beta	& 0.60	&0.76&	0.97&	1.21&	1.56&	0.63&	0.81&	0.97&	1.17&	1.49 \\
Alpha	& 2.57	&1.95&	0.18&	-1.92&	-4.00&	2.94&	1.45&	0.34&	-1.43&	-4.34 \\ \hline

\end{tabular}

\caption{\small For each month, we formed portfolios by sorting all publicly traded stocks (first five columns) and the top 1,000 stocks by market capitalization (second five columns) tracked by CRSP into five equal-sized quintiles according to trailing volatility (standard deviation) for Panel A and trailing beta for Panel B. We estimated volatility and beta by using up to 60 months of trailing returns (i.e., return data starting as early as January 1963). The return on the market, $R_m$, and the risk-free rate, $R_f$, are from Ken French's data library at Dartmouth. The information ratio uses the market return for the relevant universe, all stocks in the first five columns and the top 1,000 stocks in the last five columns. Average returns are monthly averages multiplied by 12. Standard deviation and tracking error are monthly standard deviations multiplied by the square root of 12.}

\end{table}

\end{landscape}

\clearpage


Our substantive conclusions are in line with those of \cite{lowvol} in
that low volatility and low beta portfolios outperformed high
volatility and high beta portfolios, respectively.

In Panel A of Figure \ref{figure:plotMulti}, one dollar invested in
the lowest volatility portfolio in January 1968 grows to \$62.72
(\$10.66 in real terms after adjusting for inflation) by December
2008. One dollar invested in the highest volatility portfolio during
the same time period was worth \$0.40 (\$0.07 in real terms) by
December 2008. The results are similar when only considering the 1,000
largest stocks by market capitalization as shown in Panel B of
Figure \ref{figure:plotMulti}. one dollar invested in the lowest
volatility portfolio grows to \$58.70 (\$9.98 in real terms) by
December 2008 as compared to \$5.86 (\$1.00 in real terms) for the
highest volatility portfolio.

In Panel C of Figure \ref{figure:plotMulti}, one dollar invested in
the lowest beta portfolio in January 1968 grows to \$59.84 (\$10.17 in
real terms) by December 2008. One dollar invested in the highest beta
portfolio during the same time period was worth \$6.25 (\$1.06 in real
term's) by December 2008. In Panel D of Figure \ref{figure:plotMulti},
only the 1,000 largest stocks are considered. One dollar invested in
the lowest beta portfolio grows to
\$72.61 (\$12.34 in real terms) by December 2008 as compared to \$5.68
(\$0.97 in real terms) for the highest beta portfolio.

\clearpage

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